From 4da5dc15db1469fcebbe4bbcd24878fbd2a891fe Mon Sep 17 00:00:00 2001
From: Katharine Hyatt <khyatt@flatironinstitute.org>
Date: Tue, 23 Dec 2025 03:08:52 -0500
Subject: [PATCH 1/8] Use TestSuite for orthnull

---
 test/amd/orthnull.jl        | 285 ------------------------------------
 test/cuda/orthnull.jl       | 264 ---------------------------------
 test/orthnull.jl            | 234 +++--------------------------
 test/runtests.jl            |  12 +-
 test/testsuite/TestSuite.jl |   1 +
 test/testsuite/orthnull.jl  | 268 +++++++++++++++++++++++++++++++++
 6 files changed, 293 insertions(+), 771 deletions(-)
 delete mode 100644 test/amd/orthnull.jl
 delete mode 100644 test/cuda/orthnull.jl
 create mode 100644 test/testsuite/orthnull.jl

diff --git a/test/amd/orthnull.jl b/test/amd/orthnull.jl
deleted file mode 100644
index 3223979c..00000000
--- a/test/amd/orthnull.jl
+++ /dev/null
@@ -1,285 +0,0 @@
-using MatrixAlgebraKit
-using Test
-using TestExtras
-using StableRNGs
-using LinearAlgebra: LinearAlgebra, I, mul!, diagm, norm
-using MatrixAlgebraKit: GPU_SVDAlgorithm, check_input, copy_input, default_svd_algorithm,
-    initialize_output, AbstractAlgorithm
-using AMDGPU
-
-# testing non-AbstractArray codepaths:
-include(joinpath("..", "linearmap.jl"))
-
-eltypes = (Float32, Float64, ComplexF32, ComplexF64)
-
-@testset "left_orth and left_null for T = $T" for T in eltypes
-    rng = StableRNG(123)
-    m = 54
-    @testset for n in (37, m, 63)
-        minmn = min(m, n)
-        A = ROCArray(randn(rng, T, m, n))
-        V, C = @constinferred left_orth(A)
-        N = @constinferred left_null(A)
-        @test V isa ROCMatrix{T} && size(V) == (m, minmn)
-        @test C isa ROCMatrix{T} && size(C) == (minmn, n)
-        @test N isa ROCMatrix{T} && size(N) == (m, m - minmn)
-        @test V * C ≈ A
-        @test isisometric(V)
-        @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N)
-        hV = collect(V)
-        hN = collect(N)
-        @test hV * hV' + hN * hN' ≈ I
-
-        M = LinearMap(A)
-        VM, CM = @constinferred left_orth(M; alg = :svd)
-        @test parent(VM) * parent(CM) ≈ A
-
-        if m > n
-            nullity = 5
-            V, C = @constinferred left_orth(A)
-            AMDGPU.@allowscalar begin
-                N = @constinferred left_null(A; trunc = (; maxnullity = nullity))
-            end
-            @test V isa ROCMatrix{T} && size(V) == (m, minmn)
-            @test C isa ROCMatrix{T} && size(C) == (minmn, n)
-            @test N isa ROCMatrix{T} && size(N) == (m, nullity)
-            @test V * C ≈ A
-            @test isisometric(V)
-            @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-            @test isisometric(N)
-        end
-
-        # passing a kind and some kwargs
-        V, C = @constinferred left_orth(A; alg = :qr, positive = true)
-        N = @constinferred left_null(A; alg = :qr, positive = true)
-        @test V isa ROCMatrix{T} && size(V) == (m, minmn)
-        @test C isa ROCMatrix{T} && size(C) == (minmn, n)
-        @test N isa ROCMatrix{T} && size(N) == (m, m - minmn)
-        @test V * C ≈ A
-        @test isisometric(V)
-        @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N)
-        hV = collect(V)
-        hN = collect(N)
-        @test hV * hV' + hN * hN' ≈ I
-
-        # passing an algorithm
-        V, C = @constinferred left_orth(A; alg = CUSOLVER_HouseholderQR())
-        N = @constinferred left_null(A; alg = :qr, positive = true)
-        @test V isa ROCMatrix{T} && size(V) == (m, minmn)
-        @test C isa ROCMatrix{T} && size(C) == (minmn, n)
-        @test N isa ROCMatrix{T} && size(N) == (m, m - minmn)
-        @test V * C ≈ A
-        @test isisometric(V)
-        @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N)
-        hV = collect(V)
-        hN = collect(N)
-        @test hV * hV' + hN * hN' ≈ I
-
-        Ac = similar(A)
-        V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C))
-        N2 = @constinferred left_null!(copy!(Ac, A), N)
-        @test V2 === V
-        @test C2 === C
-        @test N2 === N
-        @test V2 * C2 ≈ A
-        @test isisometric(V2)
-        @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N2)
-        hV2 = collect(V2)
-        hN2 = collect(N2)
-        @test hV2 * hV2' + hN2 * hN2' ≈ I
-
-        atol = eps(real(T))
-        V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); trunc = (; atol = atol))
-        AMDGPU.@allowscalar begin
-            N2 = @constinferred left_null!(copy!(Ac, A), N; trunc = (; atol = atol))
-        end
-        @test V2 * C2 ≈ A
-        @test isisometric(V2)
-        @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N2)
-        hV2 = collect(V2)
-        hN2 = collect(N2)
-        @test hV2 * hV2' + hN2 * hN2' ≈ I
-
-        rtol = eps(real(T))
-        for (trunc_orth, trunc_null) in (
-                ((; rtol = rtol), (; rtol = rtol)),
-                (trunctol(; rtol), trunctol(; rtol, keep_below = true)),
-            )
-            V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); trunc = trunc_orth)
-            AMDGPU.@allowscalar begin
-                N2 = @constinferred left_null!(copy!(Ac, A), N; trunc = trunc_null)
-            end
-            @test V2 * C2 ≈ A
-            @test isisometric(V2)
-            @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-            @test isisometric(N2)
-            hV2 = collect(V2)
-            hN2 = collect(N2)
-            @test hV2 * hV2' + hN2 * hN2' ≈ I
-        end
-
-        @testset for alg in (:qr, :polar, :svd) # explicit alg kwarg
-            m < n && alg == :polar && continue
-            V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); alg = $(QuoteNode(alg)))
-            @test V2 * C2 ≈ A
-            @test isisometric(V2)
-            if alg != :polar
-                N2 = @constinferred left_null!(copy!(Ac, A), N; alg)
-                @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(N2)
-                hV2 = collect(V2)
-                hN2 = collect(N2)
-                @test hV2 * hV2' + hN2 * hN2' ≈ I
-            end
-
-            # with alg and tol kwargs
-            if alg == :svd
-                V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); alg = $(QuoteNode(alg)), trunc = (; atol))
-                AMDGPU.@allowscalar begin
-                    N2 = @constinferred left_null!(copy!(Ac, A), N; alg, trunc = (; atol))
-                end
-                @test V2 * C2 ≈ A
-                @test isisometric(V2)
-                @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(N2)
-                hV2 = collect(V2)
-                hN2 = collect(N2)
-                @test hV2 * hV2' + hN2 * hN2' ≈ I
-
-                V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); alg = $(QuoteNode(alg)), trunc = (; rtol))
-                AMDGPU.@allowscalar begin
-                    N2 = @constinferred left_null!(copy!(Ac, A), N; alg, trunc = (; rtol))
-                end
-                @test V2 * C2 ≈ A
-                @test isisometric(V2)
-                @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(N2)
-                hV2 = collect(V2)
-                hN2 = collect(N2)
-                @test hV2 * hV2' + hN2 * hN2' ≈ I
-            else
-                @test_throws ArgumentError left_orth!(copy!(Ac, A), (V, C); alg, trunc = (; atol))
-                @test_throws ArgumentError left_orth!(copy!(Ac, A), (V, C); alg, trunc = (; rtol))
-                alg == :polar && continue
-                @test_throws ArgumentError left_null!(copy!(Ac, A), N; alg, trunc = (; atol))
-                @test_throws ArgumentError left_null!(copy!(Ac, A), N; alg, trunc = (; rtol))
-            end
-        end
-    end
-end
-
-@testset "right_orth and right_null for T = $T" for T in eltypes
-    rng = StableRNG(123)
-    m = 54
-    @testset for n in (37, m, 63)
-        minmn = min(m, n)
-        A = ROCArray(randn(rng, T, m, n))
-        C, Vᴴ = @constinferred right_orth(A)
-        Nᴴ = @constinferred right_null(A)
-        @test C isa ROCMatrix{T} && size(C) == (m, minmn)
-        @test Vᴴ isa ROCMatrix{T} && size(Vᴴ) == (minmn, n)
-        @test Nᴴ isa ROCMatrix{T} && size(Nᴴ) == (n - minmn, n)
-        @test C * Vᴴ ≈ A
-        @test isisometric(Vᴴ; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(Nᴴ; side = :right)
-        hVᴴ = collect(Vᴴ)
-        hNᴴ = collect(Nᴴ)
-        @test hVᴴ' * hVᴴ + hNᴴ' * hNᴴ ≈ I
-
-        M = LinearMap(A)
-        CM, VMᴴ = @constinferred right_orth(M; alg = :svd)
-        @test parent(CM) * parent(VMᴴ) ≈ A
-
-        Ac = similar(A)
-        C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ))
-        Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ)
-        @test C2 * Vᴴ2 ≈ A
-        @test isisometric(Vᴴ2; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(Nᴴ; side = :right)
-        hVᴴ2 = collect(Vᴴ2)
-        hNᴴ2 = collect(Nᴴ2)
-        @test hVᴴ2' * hVᴴ2 + hNᴴ2' * hNᴴ2 ≈ I
-
-        atol = eps(real(T))
-        rtol = eps(real(T))
-        C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); trunc = (; atol = atol))
-        AMDGPU.@allowscalar begin
-            Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; trunc = (; atol = atol))
-        end
-        @test C2 * Vᴴ2 ≈ A
-        @test isisometric(Vᴴ2; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(Nᴴ; side = :right)
-        hVᴴ2 = collect(Vᴴ2)
-        hNᴴ2 = collect(Nᴴ2)
-        @test hVᴴ2' * hVᴴ2 + hNᴴ2' * hNᴴ2 ≈ I
-
-        C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); trunc = (; rtol = rtol))
-        AMDGPU.@allowscalar begin
-            Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; trunc = (; rtol = rtol))
-        end
-        @test C2 * Vᴴ2 ≈ A
-        @test isisometric(Vᴴ2; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(Nᴴ2; side = :right)
-        hVᴴ2 = collect(Vᴴ2)
-        hNᴴ2 = collect(Nᴴ2)
-        @test hVᴴ2' * hVᴴ2 + hNᴴ2' * hNᴴ2 ≈ I
-
-
-        @testset "alg = $alg" for alg in (:lq, :polar, :svd)
-            n < m && alg == :polar && continue
-            C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = $(QuoteNode(alg)))
-            @test C2 * Vᴴ2 ≈ A
-            @test isisometric(Vᴴ2; side = :right)
-            if alg != :polar
-                Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; alg = $(QuoteNode(alg)))
-                @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(Nᴴ2; side = :right)
-                hVᴴ2 = collect(Vᴴ2)
-                hNᴴ2 = collect(Nᴴ2)
-                @test hVᴴ2' * hVᴴ2 + hNᴴ2' * hNᴴ2 ≈ I
-            end
-
-            if alg == :svd
-                C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = $(QuoteNode(alg)), trunc = (; atol))
-                AMDGPU.@allowscalar begin
-                    Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; alg = $(QuoteNode(alg)), trunc = (; atol))
-                end
-                @test C2 * Vᴴ2 ≈ A
-                @test isisometric(Vᴴ2; side = :right)
-                @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(Nᴴ2; side = :right)
-                hVᴴ2 = collect(Vᴴ2)
-                hNᴴ2 = collect(Nᴴ2)
-                @test hVᴴ2' * hVᴴ2 + hNᴴ2' * hNᴴ2 ≈ I
-
-                C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = $(QuoteNode(alg)), trunc = (; rtol))
-                AMDGPU.@allowscalar begin
-                    Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; alg = $(QuoteNode(alg)), trunc = (; rtol))
-                end
-                @test C2 * Vᴴ2 ≈ A
-                @test isisometric(Vᴴ2; side = :right)
-                @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(Nᴴ2; side = :right)
-                hVᴴ2 = collect(Vᴴ2)
-                hNᴴ2 = collect(Nᴴ2)
-                @test hVᴴ2' * hVᴴ2 + hNᴴ2' * hNᴴ2 ≈ I
-            else
-                @test_throws ArgumentError right_orth!(copy!(Ac, A), (C, Vᴴ); alg, trunc = (; atol))
-                @test_throws ArgumentError right_orth!(copy!(Ac, A), (C, Vᴴ); alg, trunc = (; rtol))
-                alg == :polar && continue
-                @test_throws ArgumentError right_null!(copy!(Ac, A), Nᴴ; alg, trunc = (; atol))
-                @test_throws ArgumentError right_null!(copy!(Ac, A), Nᴴ; alg, trunc = (; rtol))
-            end
-        end
-
-    end
-end
diff --git a/test/cuda/orthnull.jl b/test/cuda/orthnull.jl
deleted file mode 100644
index 2a2a26f6..00000000
--- a/test/cuda/orthnull.jl
+++ /dev/null
@@ -1,264 +0,0 @@
-using MatrixAlgebraKit
-using Test
-using TestExtras
-using StableRNGs
-using LinearAlgebra: LinearAlgebra, I, mul!, diagm, norm
-using MatrixAlgebraKit: GPU_SVDAlgorithm, check_input, copy_input, default_svd_algorithm,
-    initialize_output, AbstractAlgorithm
-using CUDA
-
-# testing non-AbstractArray codepaths:
-include(joinpath("..", "linearmap.jl"))
-
-eltypes = (Float32, Float64, ComplexF32, ComplexF64)
-
-@testset "left_orth and left_null for T = $T" for T in eltypes
-    rng = StableRNG(123)
-    m = 54
-    @testset for n in (37, m, 63)
-        minmn = min(m, n)
-        A = CuArray(randn(rng, T, m, n))
-        V, C = @constinferred left_orth(A)
-        N = @constinferred left_null(A)
-        @test V isa CuMatrix{T} && size(V) == (m, minmn)
-        @test C isa CuMatrix{T} && size(C) == (minmn, n)
-        @test N isa CuMatrix{T} && size(N) == (m, m - minmn)
-        @test V * C ≈ A
-        @test isisometric(V)
-        @test norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N)
-        hV = collect(V)
-        hN = collect(N)
-        @test hV * hV' + hN * hN' ≈ I
-
-        M = LinearMap(A)
-        VM, CM = @constinferred left_orth(M; alg = :svd)
-        @test parent(VM) * parent(CM) ≈ A
-
-        if m > n
-            nullity = 5
-            V, C = @constinferred left_orth(A)
-            CUDA.@allowscalar begin
-                N = @constinferred left_null(A; trunc = (; maxnullity = nullity))
-            end
-            @test V isa CuMatrix{T} && size(V) == (m, minmn)
-            @test C isa CuMatrix{T} && size(C) == (minmn, n)
-            @test N isa CuMatrix{T} && size(N) == (m, nullity)
-            @test V * C ≈ A
-            @test isisometric(V)
-            @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-            @test isisometric(N)
-        end
-
-        # passing a kind and some kwargs
-        V, C = @constinferred left_orth(A; alg = :qr, positive = true)
-        N = @constinferred left_null(A; alg = :qr, positive = true)
-        @test V isa CuMatrix{T} && size(V) == (m, minmn)
-        @test C isa CuMatrix{T} && size(C) == (minmn, n)
-        @test N isa CuMatrix{T} && size(N) == (m, m - minmn)
-        @test V * C ≈ A
-        @test isisometric(V)
-        @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N)
-        hV = collect(V)
-        hN = collect(N)
-        @test hV * hV' + hN * hN' ≈ I
-
-        # passing an algorithm
-        V, C = @constinferred left_orth(A; alg = CUSOLVER_HouseholderQR())
-        N = @constinferred left_null(A; alg = :qr, positive = true)
-        @test V isa CuMatrix{T} && size(V) == (m, minmn)
-        @test C isa CuMatrix{T} && size(C) == (minmn, n)
-        @test N isa CuMatrix{T} && size(N) == (m, m - minmn)
-        @test V * C ≈ A
-        @test isisometric(V)
-        @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N)
-        hV = collect(V)
-        hN = collect(N)
-        @test hV * hV' + hN * hN' ≈ I
-
-        Ac = similar(A)
-        V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C))
-        N2 = @constinferred left_null!(copy!(Ac, A), N)
-        @test V2 * C2 ≈ A
-        @test isisometric(V2)
-        @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N2)
-        hV2 = collect(V2)
-        hN2 = collect(N2)
-        @test hV2 * hV2' + hN2 * hN2' ≈ I
-
-        atol = eps(real(T))
-        V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); trunc = (; atol = atol))
-        N2 = @constinferred left_null!(copy!(Ac, A), N; trunc = (; atol = atol))
-        @test V2 * C2 ≈ A
-        @test isisometric(V2)
-        @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N2)
-        hV2 = collect(V2)
-        hN2 = collect(N2)
-        @test hV2 * hV2' + hN2 * hN2' ≈ I
-
-        rtol = eps(real(T))
-        for (trunc_orth, trunc_null) in (
-                ((; rtol = rtol), (; rtol = rtol)),
-                (trunctol(; rtol), trunctol(; rtol, keep_below = true)),
-            )
-            V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); trunc = trunc_orth)
-            N2 = @constinferred left_null!(copy!(Ac, A), N; trunc = trunc_null)
-            @test V2 * C2 ≈ A
-            @test isisometric(V2)
-            @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-            @test isisometric(N2)
-            hV2 = collect(V2)
-            hN2 = collect(N2)
-            @test hV2 * hV2' + hN2 * hN2' ≈ I
-        end
-
-        @testset for alg in (:qr, :polar, :svd) # explicit alg kwarg
-            m < n && alg == :polar && continue
-            V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); alg = $(QuoteNode(alg)))
-            @test V2 * C2 ≈ A
-            @test isisometric(V2)
-            if alg != :polar
-                N2 = @constinferred left_null!(copy!(Ac, A), N; alg)
-                @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(N2)
-                hV2 = collect(V2)
-                hN2 = collect(N2)
-                @test hV2 * hV2' + hN2 * hN2' ≈ I
-            end
-
-            # with alg and tol kwargs
-            if alg == :svd
-                V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); alg = $(QuoteNode(alg)), trunc = (; atol))
-                N2 = @constinferred left_null!(copy!(Ac, A), N; alg, trunc = (; atol))
-                @test V2 * C2 ≈ A
-                @test isisometric(V2)
-                @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(N2)
-                hV2 = collect(V2)
-                hN2 = collect(N2)
-                @test hV2 * hV2' + hN2 * hN2' ≈ I
-
-                V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); alg = $(QuoteNode(alg)), trunc = (; rtol))
-                N2 = @constinferred left_null!(copy!(Ac, A), N; alg, trunc = (; rtol))
-                @test V2 * C2 ≈ A
-                @test isisometric(V2)
-                @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(N2)
-                hV2 = collect(V2)
-                hN2 = collect(N2)
-                @test hV2 * hV2' + hN2 * hN2' ≈ I
-            else
-                @test_throws ArgumentError left_orth!(copy!(Ac, A), (V, C); alg, trunc = (; atol))
-                @test_throws ArgumentError left_orth!(copy!(Ac, A), (V, C); alg, trunc = (; rtol))
-                alg == :polar && continue
-                @test_throws ArgumentError left_null!(copy!(Ac, A), N; alg, trunc = (; atol))
-                @test_throws ArgumentError left_null!(copy!(Ac, A), N; alg, trunc = (; rtol))
-            end
-        end
-    end
-end
-
-@testset "right_orth and right_null for T = $T" for T in eltypes
-    rng = StableRNG(123)
-    m = 54
-    @testset for n in (37, m, 63)
-        minmn = min(m, n)
-        A = CuArray(randn(rng, T, m, n))
-        C, Vᴴ = @constinferred right_orth(A)
-        Nᴴ = @constinferred right_null(A)
-        @test C isa CuMatrix{T} && size(C) == (m, minmn)
-        @test Vᴴ isa CuMatrix{T} && size(Vᴴ) == (minmn, n)
-        @test Nᴴ isa CuMatrix{T} && size(Nᴴ) == (n - minmn, n)
-        @test C * Vᴴ ≈ A
-        @test isisometric(Vᴴ; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(Nᴴ; side = :right)
-        hVᴴ = collect(Vᴴ)
-        hNᴴ = collect(Nᴴ)
-        @test hVᴴ' * hVᴴ + hNᴴ' * hNᴴ ≈ I
-
-        M = LinearMap(A)
-        CM, VMᴴ = @constinferred right_orth(M; alg = :svd)
-        @test parent(CM) * parent(VMᴴ) ≈ A
-
-        Ac = similar(A)
-        C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ))
-        Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ)
-        @test C2 * Vᴴ2 ≈ A
-        @test isisometric(Vᴴ2; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(Nᴴ; side = :right)
-        hVᴴ2 = collect(Vᴴ2)
-        hNᴴ2 = collect(Nᴴ2)
-        @test hVᴴ2' * hVᴴ2 + hNᴴ2' * hNᴴ2 ≈ I
-
-        atol = eps(real(T))
-        rtol = eps(real(T))
-        C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); trunc = (; atol = atol))
-        Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; trunc = (; atol = atol))
-        @test C2 * Vᴴ2 ≈ A
-        @test isisometric(Vᴴ2; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(Nᴴ; side = :right)
-        hVᴴ2 = collect(Vᴴ2)
-        hNᴴ2 = collect(Nᴴ2)
-        @test hVᴴ2' * hVᴴ2 + hNᴴ2' * hNᴴ2 ≈ I
-
-        C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); trunc = (; rtol = rtol))
-        Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; trunc = (; rtol = rtol))
-        @test C2 * Vᴴ2 ≈ A
-        @test isisometric(Vᴴ2; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(Nᴴ2; side = :right)
-        hVᴴ2 = collect(Vᴴ2)
-        hNᴴ2 = collect(Nᴴ2)
-        @test hVᴴ2' * hVᴴ2 + hNᴴ2' * hNᴴ2 ≈ I
-
-        @testset "alg = $alg" for alg in (:lq, :polar, :svd)
-            n < m && alg == :polar && continue
-            C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = $(QuoteNode(alg)))
-            @test C2 * Vᴴ2 ≈ A
-            @test isisometric(Vᴴ2; side = :right)
-            if alg != :polar
-                Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; alg = $(QuoteNode(alg)))
-                @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(Nᴴ2; side = :right)
-                hVᴴ2 = collect(Vᴴ2)
-                hNᴴ2 = collect(Nᴴ2)
-                @test hVᴴ2' * hVᴴ2 + hNᴴ2' * hNᴴ2 ≈ I
-            end
-
-            if alg == :svd
-                C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = $(QuoteNode(alg)), trunc = (; atol))
-                Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; alg = $(QuoteNode(alg)), trunc = (; atol))
-                @test C2 * Vᴴ2 ≈ A
-                @test isisometric(Vᴴ2; side = :right)
-                @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(Nᴴ2; side = :right)
-                hVᴴ2 = collect(Vᴴ2)
-                hNᴴ2 = collect(Nᴴ2)
-                @test hVᴴ2' * hVᴴ2 + hNᴴ2' * hNᴴ2 ≈ I
-
-                C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = $(QuoteNode(alg)), trunc = (; rtol))
-                Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; alg = $(QuoteNode(alg)), trunc = (; rtol))
-                @test C2 * Vᴴ2 ≈ A
-                @test isisometric(Vᴴ2; side = :right)
-                @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(Nᴴ2; side = :right)
-                hVᴴ2 = collect(Vᴴ2)
-                hNᴴ2 = collect(Nᴴ2)
-                @test hVᴴ2' * hVᴴ2 + hNᴴ2' * hNᴴ2 ≈ I
-            else
-                @test_throws ArgumentError right_orth!(copy!(Ac, A), (C, Vᴴ); alg, trunc = (; atol))
-                @test_throws ArgumentError right_orth!(copy!(Ac, A), (C, Vᴴ); alg, trunc = (; rtol))
-                alg == :polar && continue
-                @test_throws ArgumentError right_null!(copy!(Ac, A), Nᴴ; alg, trunc = (; atol))
-                @test_throws ArgumentError right_null!(copy!(Ac, A), Nᴴ; alg, trunc = (; rtol))
-            end
-        end
-    end
-end
diff --git a/test/orthnull.jl b/test/orthnull.jl
index ce742e8f..874971ec 100644
--- a/test/orthnull.jl
+++ b/test/orthnull.jl
@@ -2,225 +2,33 @@ using MatrixAlgebraKit
 using Test
 using TestExtras
 using StableRNGs
-using LinearAlgebra: LinearAlgebra, I
+using LinearAlgebra: LinearAlgebra, I, Diagonal
+using CUDA, AMDGPU
 
-# testing non-AbstractArray codepaths:
-include("linearmap.jl")
+BLASFloats = (Float32, Float64, ComplexF32, ComplexF64)
+GenericFloats = (Float16, BigFloat, Complex{BigFloat})
 
-eltypes = (Float32, Float64, ComplexF32, ComplexF64)
-@testset "left_orth and left_null for T = $T" for T in eltypes
-    rng = StableRNG(123)
-    m = 54
-    for n in (37, m, 63)
-        minmn = min(m, n)
-        A = randn(rng, T, m, n)
-        V, C = @constinferred left_orth(A)
-        N = @constinferred left_null(A)
-        @test V isa Matrix{T} && size(V) == (m, minmn)
-        @test C isa Matrix{T} && size(C) == (minmn, n)
-        @test N isa Matrix{T} && size(N) == (m, m - minmn)
-        @test V * C ≈ A
-        @test isisometric(V)
-        @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N)
-        @test V * V' + N * N' ≈ I
+@isdefined(TestSuite) || include("testsuite/TestSuite.jl")
+using .TestSuite
 
-        M = LinearMap(A)
-        VM, CM = @constinferred left_orth(M; alg = :svd)
-        @test parent(VM) * parent(CM) ≈ A
+is_buildkite = get(ENV, "BUILDKITE", "false") == "true"
 
-        if m > n
-            nullity = 5
-            V, C = @constinferred left_orth(A)
-            N = @constinferred left_null(A; trunc = (; maxnullity = nullity))
-            @test V isa Matrix{T} && size(V) == (m, minmn)
-            @test C isa Matrix{T} && size(C) == (minmn, n)
-            @test N isa Matrix{T} && size(N) == (m, nullity)
-            @test V * C ≈ A
-            @test isisometric(V)
-            @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-            @test isisometric(N)
+m = 54
+for T in (BLASFloats..., GenericFloats...), n in (37, m, 63)
+    TestSuite.seed_rng!(123)
+    if T ∈ BLASFloats
+        if CUDA.functional()
+            TestSuite.test_orthnull(CuMatrix{T}, (m, n))
+            n == m && TestSuite.test_orthnull(Diagonal{T, CuVector{T}}, m)
         end
-
-        # passing a kind and some kwargs
-        V, C = @constinferred left_orth(A; alg = :qr, positive = true)
-        N = @constinferred left_null(A; alg = :qr, positive = true)
-        @test V isa Matrix{T} && size(V) == (m, minmn)
-        @test C isa Matrix{T} && size(C) == (minmn, n)
-        @test N isa Matrix{T} && size(N) == (m, m - minmn)
-        @test V * C ≈ A
-        @test isisometric(V)
-        @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N)
-        @test V * V' + N * N' ≈ I
-
-        # passing an algorithm
-        V, C = @constinferred left_orth(A; alg = LAPACK_HouseholderQR())
-        N = @constinferred left_null(A; alg = :qr, positive = true)
-        @test V isa Matrix{T} && size(V) == (m, minmn)
-        @test C isa Matrix{T} && size(C) == (minmn, n)
-        @test N isa Matrix{T} && size(N) == (m, m - minmn)
-        @test V * C ≈ A
-        @test isisometric(V)
-        @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N)
-        @test V * V' + N * N' ≈ I
-
-        Ac = similar(A)
-        V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C))
-        N2 = @constinferred left_null!(copy!(Ac, A), N)
-        @test V2 * C2 ≈ A
-        @test isisometric(V2)
-        @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N2)
-        @test V2 * V2' + N2 * N2' ≈ I
-
-        atol = eps(real(T))
-        V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); trunc = (; atol = atol))
-        N2 = @constinferred left_null!(copy!(Ac, A), N; trunc = (; atol = atol))
-        @test V2 * C2 ≈ A
-        @test isisometric(V2)
-        @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N2)
-        @test V2 * V2' + N2 * N2' ≈ I
-
-        rtol = eps(real(T))
-        for (trunc_orth, trunc_null) in (
-                ((; rtol = rtol), (; rtol = rtol)),
-                (trunctol(; rtol), trunctol(; rtol, keep_below = true)),
-            )
-            V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); trunc = trunc_orth)
-            N2 = @constinferred left_null!(copy!(Ac, A), N; trunc = trunc_null)
-            @test V2 * C2 ≈ A
-            @test isisometric(V2)
-            @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-            @test isisometric(N2)
-            @test V2 * V2' + N2 * N2' ≈ I
-        end
-
-        for alg in (:qr, :polar, :svd) # explicit kind kwarg
-            m < n && alg === :polar && continue
-            V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); alg = $(QuoteNode(alg)))
-            @test V2 * C2 ≈ A
-            @test isisometric(V2)
-            if alg != :polar
-                N2 = @constinferred left_null!(copy!(Ac, A), N; alg)
-                @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(N2)
-                @test V2 * V2' + N2 * N2' ≈ I
-            end
-
-            # with kind and tol kwargs
-            if alg == :svd
-                V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); alg = $(QuoteNode(alg)), trunc = (; atol))
-                N2 = @constinferred left_null!(copy!(Ac, A), N; alg, trunc = (; atol))
-                @test V2 * C2 ≈ A
-                @test isisometric(V2)
-                @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(N2)
-                @test V2 * V2' + N2 * N2' ≈ I
-
-                V2, C2 = @constinferred left_orth!(copy!(Ac, A), (V, C); alg = $(QuoteNode(alg)), trunc = (; rtol))
-                N2 = @constinferred left_null!(copy!(Ac, A), N; alg, trunc = (; rtol))
-                @test V2 * C2 ≈ A
-                @test isisometric(V2)
-                @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(N2)
-                @test V2 * V2' + N2 * N2' ≈ I
-            else
-                @test_throws ArgumentError left_orth!(copy!(Ac, A), (V, C); alg, trunc = (; atol))
-                @test_throws ArgumentError left_orth!(copy!(Ac, A), (V, C); alg, trunc = (; rtol))
-                alg == :polar && continue
-                @test_throws ArgumentError left_null!(copy!(Ac, A), N; alg, trunc = (; atol))
-                @test_throws ArgumentError left_null!(copy!(Ac, A), N; alg, trunc = (; rtol))
-            end
+        if AMDGPU.functional()
+            TestSuite.test_orthnull(ROCMatrix{T}, (m, n))
+            n == m && TestSuite.test_orthnull(Diagonal{T, ROCVector{T}}, m)
         end
     end
-end
-
-@testset "right_orth and right_null for T = $T" for T in eltypes
-    rng = StableRNG(123)
-    m = 54
-    for n in (37, m, 63)
-        minmn = min(m, n)
-        A = randn(rng, T, m, n)
-        C, Vᴴ = @constinferred right_orth(A)
-        Nᴴ = @constinferred right_null(A)
-        @test C isa Matrix{T} && size(C) == (m, minmn)
-        @test Vᴴ isa Matrix{T} && size(Vᴴ) == (minmn, n)
-        @test Nᴴ isa Matrix{T} && size(Nᴴ) == (n - minmn, n)
-        @test C * Vᴴ ≈ A
-        @test isisometric(Vᴴ; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(Nᴴ; side = :right)
-        @test Vᴴ' * Vᴴ + Nᴴ' * Nᴴ ≈ I
-
-        M = LinearMap(A)
-        CM, VMᴴ = @constinferred right_orth(M; alg = :svd)
-        @test parent(CM) * parent(VMᴴ) ≈ A
-
-        Ac = similar(A)
-        C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ))
-        Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ)
-        @test C2 * Vᴴ2 ≈ A
-        @test isisometric(Vᴴ2; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(Nᴴ; side = :right)
-        @test Vᴴ2' * Vᴴ2 + Nᴴ2' * Nᴴ2 ≈ I
-
-        atol = eps(real(T))
-        C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); trunc = (; atol))
-        Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; trunc = (; atol))
-        @test C2 * Vᴴ2 ≈ A
-        @test isisometric(Vᴴ2; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(Nᴴ; side = :right)
-        @test Vᴴ2' * Vᴴ2 + Nᴴ2' * Nᴴ2 ≈ I
-
-        rtol = eps(real(T))
-        C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); trunc = (; rtol))
-        Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; trunc = (; rtol))
-        @test C2 * Vᴴ2 ≈ A
-        @test isisometric(Vᴴ2; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(Nᴴ2; side = :right)
-        @test Vᴴ2' * Vᴴ2 + Nᴴ2' * Nᴴ2 ≈ I
-
-        for alg in (:lq, :polar, :svd)
-            n < m && alg == :polar && continue
-            C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = $(QuoteNode(alg)))
-            @test C2 * Vᴴ2 ≈ A
-            @test isisometric(Vᴴ2; side = :right)
-            if alg != :polar
-                Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; alg = $(QuoteNode(alg)))
-                @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(Nᴴ2; side = :right)
-                @test Vᴴ2' * Vᴴ2 + Nᴴ2' * Nᴴ2 ≈ I
-            end
-
-            if alg == :svd
-                C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = $(QuoteNode(alg)), trunc = (; atol))
-                Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; alg = $(QuoteNode(alg)), trunc = (; atol))
-                @test C2 * Vᴴ2 ≈ A
-                @test isisometric(Vᴴ2; side = :right)
-                @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(Nᴴ2; side = :right)
-                @test Vᴴ2' * Vᴴ2 + Nᴴ2' * Nᴴ2 ≈ I
-
-                C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = $(QuoteNode(alg)), trunc = (; rtol))
-                Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; alg = $(QuoteNode(alg)), trunc = (; rtol))
-                @test C2 * Vᴴ2 ≈ A
-                @test isisometric(Vᴴ2; side = :right)
-                @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(Nᴴ2; side = :right)
-                @test Vᴴ2' * Vᴴ2 + Nᴴ2' * Nᴴ2 ≈ I
-            else
-                @test_throws ArgumentError right_orth!(copy!(Ac, A), (C, Vᴴ); alg, trunc = (; atol))
-                @test_throws ArgumentError right_orth!(copy!(Ac, A), (C, Vᴴ); alg, trunc = (; rtol))
-                alg == :polar && continue
-                @test_throws ArgumentError right_null!(copy!(Ac, A), Nᴴ; alg, trunc = (; atol))
-                @test_throws ArgumentError right_null!(copy!(Ac, A), Nᴴ; alg, trunc = (; rtol))
-            end
-        end
+    if !is_buildkite
+        TestSuite.test_orthnull(T, (m, n))
+        AT = Diagonal{T, Vector{T}}
+        TestSuite.test_orthnull(AT, m)
     end
 end
diff --git a/test/runtests.jl b/test/runtests.jl
index baeb77bb..9fe64002 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -16,9 +16,6 @@ if !is_buildkite
     @safetestset "Generalized Eigenvalue Decomposition" begin
         include("gen_eig.jl")
     end
-    @safetestset "Image and Null Space" begin
-        include("orthnull.jl")
-    end
     @safetestset "Mooncake" begin
         include("mooncake.jl")
     end
@@ -63,15 +60,15 @@ end
 @safetestset "Hermitian Eigenvalue Decomposition" begin
     include("eigh.jl")
 end
+@safetestset "Image and Null Space" begin
+    include("orthnull.jl")
+end
 
 using CUDA
 if CUDA.functional()
     @safetestset "CUDA SVD" begin
         include("cuda/svd.jl")
     end
-    @safetestset "CUDA Image and Null Space" begin
-        include("cuda/orthnull.jl")
-    end
 end
 
 using AMDGPU
@@ -79,7 +76,4 @@ if AMDGPU.functional()
     @safetestset "AMDGPU SVD" begin
         include("amd/svd.jl")
     end
-    @safetestset "AMDGPU Image and Null Space" begin
-        include("amd/orthnull.jl")
-    end
 end
diff --git a/test/testsuite/TestSuite.jl b/test/testsuite/TestSuite.jl
index 25da777a..f288eff5 100644
--- a/test/testsuite/TestSuite.jl
+++ b/test/testsuite/TestSuite.jl
@@ -76,5 +76,6 @@ include("projections.jl")
 include("schur.jl")
 include("eig.jl")
 include("eigh.jl")
+include("orthnull.jl")
 
 end
diff --git a/test/testsuite/orthnull.jl b/test/testsuite/orthnull.jl
new file mode 100644
index 00000000..fa09214e
--- /dev/null
+++ b/test/testsuite/orthnull.jl
@@ -0,0 +1,268 @@
+using TestExtras
+using LinearAlgebra
+
+include("../linearmap.jl")
+
+function test_orthnull(T::Type, sz; kwargs...)
+    summary_str = testargs_summary(T, sz)
+    return @testset "orthnull $summary_str" begin
+        test_left_orthnull(T, sz; kwargs...)
+        test_right_orthnull(T, sz; kwargs...)
+    end
+end
+
+function test_left_orthnull(
+        T::Type, sz;
+        atol::Real = 0, rtol::Real = precision(T),
+        kwargs...
+    )
+    summary_str = testargs_summary(T, sz)
+    return @testset "left_orth! and left_null! $summary_str" begin
+        A = instantiate_matrix(T, sz)
+        Ac = deepcopy(A)
+        V, C = @testinferred left_orth(A)
+        N = @testinferred left_null(A)
+        m, n = size(A)
+        minmn = min(m, n)
+        @test V isa typeof(A) && size(V) == (m, minmn)
+        @test C isa typeof(A) && size(C) == (minmn, n)
+        @test eltype(N) == eltype(A) && size(N) == (m, m - minmn)
+        @test V * C ≈ A
+        @test isisometric(V)
+        @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+        @test isisometric(N)
+        @test collect(V) * collect(V)' + collect(N) * collect(N)' ≈ I
+
+        M = LinearMap(A)
+        # broken
+        #VM, CM = @testinferred left_orth(M; alg = :svd)
+        VM, CM = left_orth(M; alg = :svd)
+        @test parent(VM) * parent(CM) ≈ A
+
+        if m > n && (T <: Number  || T <: Diagonal{<:Number, <:Vector})
+            nullity = 5
+            V, C = @testinferred left_orth(A)
+            N = @testinferred left_null(A; trunc = (; maxnullity = nullity))
+            @test V isa typeof(A) && size(V) == (m, minmn)
+            @test C isa typeof(A) && size(C) == (minmn, n)
+            @test eltype(N) == eltype(A) && size(N) == (m, nullity)
+            @test V * C ≈ A
+            @test isisometric(V)
+            @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+            @test isisometric(N)
+        end
+
+        # passing a kind and some kwargs
+        # broken
+        # V, C = @testinferred left_orth(A; alg = :qr, positive = true)
+        V, C = left_orth(A; alg = :qr, positive = true)
+        N = @testinferred left_null(A; alg = :qr, positive = true)
+        @test V isa typeof(A) && size(V) == (m, minmn)
+        @test C isa typeof(A) && size(C) == (minmn, n)
+        @test eltype(N) == eltype(A) && size(N) == (m, m - minmn)
+        @test V * C ≈ A
+        @test isisometric(V)
+        @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+        @test isisometric(N)
+        @test collect(V) * collect(V)' + collect(N) * collect(N)' ≈ I
+
+        # passing an algorithm
+        if !isa(A, Diagonal)
+            V, C = @testinferred left_orth(A; alg = MatrixAlgebraKit.default_qr_algorithm(A))
+            N = @testinferred left_null(A; alg = :qr, positive = true)
+            @test V isa typeof(A) && size(V) == (m, minmn)
+            @test C isa typeof(A) && size(C) == (minmn, n)
+            @test eltype(N) == eltype(A) && size(N) == (m, m - minmn)
+            @test V * C ≈ A
+            @test isisometric(V)
+            @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+            @test isisometric(N)
+            @test collect(V) * collect(V)' + collect(N) * collect(N)' ≈ I
+        end
+
+        Ac = similar(A)
+        V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C))
+        N2 = @testinferred left_null!(copy!(Ac, A), N)
+        @test V2 * C2 ≈ A
+        @test isisometric(V2)
+        @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+        @test isisometric(N2)
+        @test collect(V2) * collect(V2)' + collect(N2) * collect(N2)' ≈ I
+
+        # doesn't work on AMD...
+        atol = eps(real(eltype(T)))
+        V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C); trunc = (; atol = atol))
+        N2 = @testinferred left_null!(copy!(Ac, A), N; trunc = (; atol = atol))
+        @test V2 * C2 ≈ A
+        @test isisometric(V2)
+        @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+        @test isisometric(N2)
+        @test collect(V2) * collect(V2)' + collect(N2) * collect(N2)' ≈ I
+
+        if (T <: Number  || T <: Diagonal{<:Number, <:Vector})
+            rtol = eps(real(eltype(T)))
+            for (trunc_orth, trunc_null) in (
+                    ((; rtol = rtol), (; rtol = rtol)),
+                    (trunctol(; rtol), trunctol(; rtol, keep_below = true)),
+                )
+                V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C); trunc = trunc_orth)
+                N2 = @testinferred left_null!(copy!(Ac, A), N; trunc = trunc_null)
+                @test V2 * C2 ≈ A
+                @test isisometric(V2)
+                @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+                @test isisometric(N2)
+                @test collect(V2) * collect(V2)' + collect(N2) * collect(N2)' ≈ I
+            end
+        end
+
+        for alg in (:qr, :polar, :svd) # explicit kind kwarg
+            m < n && alg === :polar && continue
+            # broken
+            # V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C); alg = alg)
+            V2, C2 = left_orth!(copy!(Ac, A), (V, C); alg = alg)
+            @test V2 * C2 ≈ A
+            @test isisometric(V2)
+            if alg != :polar
+                N2 = @testinferred left_null!(copy!(Ac, A), N; alg)
+                @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+                @test isisometric(N2)
+                @test collect(V2) * collect(V2)' + collect(N2) * collect(N2)' ≈ I
+            end
+
+            # with kind and tol kwargs
+            if alg == :svd
+                if (T <: Number  || T <: Diagonal{<:Number, <:Vector})
+                    # broken
+                    # V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C); alg = alg, trunc = (; atol))
+                    V2, C2 = left_orth!(copy!(Ac, A), (V, C); alg = alg, trunc = (; atol))
+                    N2 = @testinferred left_null!(copy!(Ac, A), N; alg, trunc = (; atol))
+                    @test V2 * C2 ≈ A
+                    @test isisometric(V2)
+                    @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+                    @test isisometric(N2)
+                    @test collect(V2) * collect(V2)' + collect(N2) * collect(N2)' ≈ I
+
+                    # broken
+                    # V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C); alg = alg, trunc = (; rtol))
+                    V2, C2 = left_orth!(copy!(Ac, A), (V, C); alg = alg, trunc = (; rtol))
+                    N2 = @testinferred left_null!(copy!(Ac, A), N; alg, trunc = (; rtol))
+                    @test V2 * C2 ≈ A
+                    @test isisometric(V2)
+                    @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+                    @test isisometric(N2)
+                    @test collect(V2) * collect(V2)' + collect(N2) * collect(N2)' ≈ I
+                end
+            else
+                @test_throws ArgumentError left_orth!(copy!(Ac, A), (V, C); alg, trunc = (; atol))
+                @test_throws ArgumentError left_orth!(copy!(Ac, A), (V, C); alg, trunc = (; rtol))
+                alg == :polar && continue
+                @test_throws ArgumentError left_null!(copy!(Ac, A), N; alg, trunc = (; atol))
+                @test_throws ArgumentError left_null!(copy!(Ac, A), N; alg, trunc = (; rtol))
+            end
+        end
+    end
+end
+
+function test_right_orthnull(
+        T::Type, sz;
+        atol::Real = 0, rtol::Real = precision(T),
+        kwargs...
+    )
+    summary_str = testargs_summary(T, sz)
+    return @testset "right_orth! and right_null! $summary_str" begin
+        A = instantiate_matrix(T, sz)
+        m, n = size(A)
+        minmn = min(m, n)
+        Ac = deepcopy(A)
+        C, Vᴴ = @testinferred right_orth(A)
+        Nᴴ = @testinferred right_null(A)
+        @test C isa typeof(A) && size(C) == (m, minmn)
+        @test Vᴴ isa typeof(A) && size(Vᴴ) == (minmn, n)
+        @test eltype(Nᴴ) == eltype(A) && size(Nᴴ) == (n - minmn, n)
+        @test C * Vᴴ ≈ A
+        @test isisometric(Vᴴ; side = :right)
+        @test LinearAlgebra.norm(A * adjoint(Nᴴ)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+        @test isisometric(Nᴴ; side = :right)
+        @test collect(Vᴴ)' * collect(Vᴴ) + collect(Nᴴ)' * collect(Nᴴ) ≈ I
+
+        M = LinearMap(A)
+        # broken
+        #CM, VMᴴ = @testinferred right_orth(M; alg = :svd)
+        CM, VMᴴ = right_orth(M; alg = :svd)
+        @test parent(CM) * parent(VMᴴ) ≈ A
+
+        Ac = similar(A)
+        C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ))
+        Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ)
+        @test C2 * Vᴴ2 ≈ A
+        @test isisometric(Vᴴ2; side = :right)
+        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+        @test isisometric(Nᴴ; side = :right)
+        @test collect(Vᴴ2)' * collect(Vᴴ2) + collect(Nᴴ2)' * collect(Nᴴ2) ≈ I
+
+        if (T <: Number  || T <: Diagonal{<:Number, <:Vector})
+            atol = eps(real(eltype(T)))
+            C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ); trunc = (; atol))
+            Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; trunc = (; atol))
+            @test C2 * Vᴴ2 ≈ A
+            @test isisometric(Vᴴ2; side = :right)
+            @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+            @test isisometric(Nᴴ; side = :right)
+            @test collect(Vᴴ2)' * collect(Vᴴ2) + collect(Nᴴ2)' * collect(Nᴴ2) ≈ I
+
+            rtol = eps(real(eltype(T)))
+            C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ); trunc = (; rtol))
+            Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; trunc = (; rtol))
+            @test C2 * Vᴴ2 ≈ A
+            @test isisometric(Vᴴ2; side = :right)
+            @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+            @test isisometric(Nᴴ2; side = :right)
+            @test collect(Vᴴ2)' * collect(Vᴴ2) + collect(Nᴴ2)' * collect(Nᴴ2) ≈ I
+        end
+
+        for alg in (:lq, :polar, :svd)
+            n < m && alg == :polar && continue
+            # broken
+            #C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = alg)
+            C2, Vᴴ2 = right_orth!(copy!(Ac, A), (C, Vᴴ); alg = alg)
+            @test C2 * Vᴴ2 ≈ A
+            @test isisometric(Vᴴ2; side = :right)
+            if alg != :polar
+                Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; alg = alg)
+                @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+                @test isisometric(Nᴴ2; side = :right)
+                @test collect(Vᴴ2)' * collect(Vᴴ2) + collect(Nᴴ2)' * collect(Nᴴ2) ≈ I
+            end
+
+            if alg == :svd
+                if (T <: Number  || T <: Diagonal{<:Number, <:Vector})
+                    # broken
+                    #C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = alg, trunc = (; atol))
+                    C2, Vᴴ2 = right_orth!(copy!(Ac, A), (C, Vᴴ); alg = alg, trunc = (; atol))
+                    Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; alg = alg, trunc = (; atol))
+                    @test C2 * Vᴴ2 ≈ A
+                    @test isisometric(Vᴴ2; side = :right)
+                    @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+                    @test isisometric(Nᴴ2; side = :right)
+                    @test collect(Vᴴ2)' * collect(Vᴴ2) + collect(Nᴴ2)' * collect(Nᴴ2) ≈ I
+
+                    # broken
+                    #C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = alg, trunc = (; rtol))
+                    C2, Vᴴ2 = right_orth!(copy!(Ac, A), (C, Vᴴ); alg = alg, trunc = (; rtol))
+                    Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; alg = alg, trunc = (; rtol))
+                    @test C2 * Vᴴ2 ≈ A
+                    @test isisometric(Vᴴ2; side = :right)
+                    @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+                    @test isisometric(Nᴴ2; side = :right)
+                    @test collect(Vᴴ2)' * collect(Vᴴ2) + collect(Nᴴ2)' * collect(Nᴴ2) ≈ I
+                end
+            else
+                @test_throws ArgumentError right_orth!(copy!(Ac, A), (C, Vᴴ); alg, trunc = (; atol))
+                @test_throws ArgumentError right_orth!(copy!(Ac, A), (C, Vᴴ); alg, trunc = (; rtol))
+                alg == :polar && continue
+                @test_throws ArgumentError right_null!(copy!(Ac, A), Nᴴ; alg, trunc = (; atol))
+                @test_throws ArgumentError right_null!(copy!(Ac, A), Nᴴ; alg, trunc = (; rtol))
+            end
+        end
+    end
+end

From 1eb79ef94bdc01ecfd2c362abdbdfec11f2e1262 Mon Sep 17 00:00:00 2001
From: Katharine Hyatt <kslimes@gmail.com>
Date: Tue, 23 Dec 2025 11:22:52 +0100
Subject: [PATCH 2/8] Don't test nonworking stuff

---
 test/orthnull.jl | 8 +++++---
 1 file changed, 5 insertions(+), 3 deletions(-)

diff --git a/test/orthnull.jl b/test/orthnull.jl
index 874971ec..5db9d2de 100644
--- a/test/orthnull.jl
+++ b/test/orthnull.jl
@@ -27,8 +27,10 @@ for T in (BLASFloats..., GenericFloats...), n in (37, m, 63)
         end
     end
     if !is_buildkite
-        TestSuite.test_orthnull(T, (m, n))
-        AT = Diagonal{T, Vector{T}}
-        TestSuite.test_orthnull(AT, m)
+        if T ∈ BLASFloats # no qr_null or lq_null for GenericFloats
+            TestSuite.test_orthnull(T, (m, n))
+        end
+        #AT = Diagonal{T, Vector{T}}
+        #TestSuite.test_orthnull(AT, m) # not supported
     end
 end

From 964bf1e6b9d3c0b7247b244ff87a7b20e35ab9a9 Mon Sep 17 00:00:00 2001
From: Katharine Hyatt <khyatt@flatironinstitute.org>
Date: Tue, 23 Dec 2025 08:47:16 -0500
Subject: [PATCH 3/8] Fix linearmap

---
 test/linearmap.jl | 8 +++++++-
 1 file changed, 7 insertions(+), 1 deletion(-)

diff --git a/test/linearmap.jl b/test/linearmap.jl
index bbf84e5e..ad6102d7 100644
--- a/test/linearmap.jl
+++ b/test/linearmap.jl
@@ -3,7 +3,7 @@ module LinearMaps
     export LinearMap
 
     using MatrixAlgebraKit
-    using MatrixAlgebraKit: AbstractAlgorithm
+    using MatrixAlgebraKit: AbstractAlgorithm, DiagonalAlgorithm
     import MatrixAlgebraKit as MAK
 
     using LinearAlgebra: LinearAlgebra, lmul!, rmul!
@@ -32,6 +32,12 @@ module LinearMaps
             LinearMap.(MAK.initialize_output($f!, parent(A), alg))
         @eval MAK.$f!(A::LinearMap, F, alg::AbstractAlgorithm) =
             LinearMap.(MAK.$f!(parent(A), parent.(F), alg))
+        @eval MAK.check_input(::typeof($f!), A::LinearMap, F, alg::DiagonalAlgorithm) =
+            MAK.check_input($f!, parent(A), parent.(F), alg)
+        @eval MAK.initialize_output(::typeof($f!), A::LinearMap, alg::DiagonalAlgorithm) =
+            LinearMap.(MAK.initialize_output($f!, parent(A), alg))
+        @eval MAK.$f!(A::LinearMap, F, alg::DiagonalAlgorithm) =
+            LinearMap.(MAK.$f!(parent(A), parent.(F), alg))
     end
 
     for f in (:qr, :lq, :svd)

From 9240221f93e5be06a5a686a5bbea59bd719d398a Mon Sep 17 00:00:00 2001
From: Katharine Hyatt <khyatt@flatironinstitute.org>
Date: Tue, 23 Dec 2025 09:15:31 -0500
Subject: [PATCH 4/8] Fix AMD

---
 .../MatrixAlgebraKitAMDGPUExt.jl              | 39 -------------------
 1 file changed, 39 deletions(-)

diff --git a/ext/MatrixAlgebraKitAMDGPUExt/MatrixAlgebraKitAMDGPUExt.jl b/ext/MatrixAlgebraKitAMDGPUExt/MatrixAlgebraKitAMDGPUExt.jl
index fd0c1604..abfa6353 100644
--- a/ext/MatrixAlgebraKitAMDGPUExt/MatrixAlgebraKitAMDGPUExt.jl
+++ b/ext/MatrixAlgebraKitAMDGPUExt/MatrixAlgebraKitAMDGPUExt.jl
@@ -159,45 +159,6 @@ function MatrixAlgebraKit._avgdiff!(A::StridedROCMatrix, B::StridedROCMatrix)
     return A, B
 end
 
-function MatrixAlgebraKit.truncate(
-        ::typeof(left_null!), US::Tuple{TU, TS}, strategy::TruncationStrategy
-    ) where {TU <: ROCMatrix, TS}
-    # TODO: avoid allocation?
-    U, S = US
-    extended_S = vcat(diagview(S), zeros(eltype(S), max(0, size(S, 1) - size(S, 2))))
-    ind = MatrixAlgebraKit.findtruncated(extended_S, strategy)
-    trunc_cols = collect(1:size(U, 2))[ind]
-    Utrunc = U[:, trunc_cols]
-    return Utrunc, ind
-end
-function MatrixAlgebraKit.truncate(
-        ::typeof(right_null!), SVᴴ::Tuple{TS, TVᴴ}, strategy::TruncationStrategy
-    ) where {TS, TVᴴ <: ROCMatrix}
-    # TODO: avoid allocation?
-    S, Vᴴ = SVᴴ
-    extended_S = vcat(diagview(S), zeros(eltype(S), max(0, size(S, 2) - size(S, 1))))
-    ind = MatrixAlgebraKit.findtruncated(extended_S, strategy)
-    trunc_rows = collect(1:size(Vᴴ, 1))[ind]
-    Vᴴtrunc = Vᴴ[trunc_rows, :]
-    return Vᴴtrunc, ind
-end
-
-# disambiguate:
-function MatrixAlgebraKit.truncate(
-        ::typeof(left_null!), (U, S)::Tuple{TU, TS}, ::NoTruncation
-    ) where {TU <: ROCMatrix, TS}
-    m, n = size(S)
-    ind = (n + 1):m
-    return U[:, ind], ind
-end
-function MatrixAlgebraKit.truncate(
-        ::typeof(right_null!), (S, Vᴴ)::Tuple{TS, TVᴴ}, ::NoTruncation
-    ) where {TS, TVᴴ <: ROCMatrix}
-    m, n = size(S)
-    ind = (m + 1):n
-    return Vᴴ[ind, :], ind
-end
-
 # avoids calling the BlasMat specialization that assumes syrk! or herk! is called
 # TODO: remove once syrk! or herk! is defined
 function MatrixAlgebraKit._mul_herm!(C::StridedROCMatrix{T}, A::StridedROCMatrix{T}) where {T <: BlasFloat}

From 24bc6d2dc73b8869545c4184c6416dcff1c623be Mon Sep 17 00:00:00 2001
From: Katharine Hyatt <katharine.s.hyatt@gmail.com>
Date: Sat, 27 Dec 2025 20:05:33 +0100
Subject: [PATCH 5/8] Support GenericFloats too

---
 ext/MatrixAlgebraKitGenericLinearAlgebraExt.jl | 3 +++
 test/linearmap.jl                              | 7 ++++++-
 test/orthnull.jl                               | 6 ++----
 3 files changed, 11 insertions(+), 5 deletions(-)

diff --git a/ext/MatrixAlgebraKitGenericLinearAlgebraExt.jl b/ext/MatrixAlgebraKitGenericLinearAlgebraExt.jl
index 14c63a4e..b78b38a0 100644
--- a/ext/MatrixAlgebraKitGenericLinearAlgebraExt.jl
+++ b/ext/MatrixAlgebraKitGenericLinearAlgebraExt.jl
@@ -2,6 +2,7 @@ module MatrixAlgebraKitGenericLinearAlgebraExt
 
 using MatrixAlgebraKit
 using MatrixAlgebraKit: sign_safe, check_input, diagview, gaugefix!, one!, default_fixgauge
+using MatrixAlgebraKit: left_orth_alg
 using GenericLinearAlgebra: svd!, svdvals!, eigen!, eigvals!, Hermitian, qr!
 using LinearAlgebra: I, Diagonal, lmul!
 
@@ -133,4 +134,6 @@ function MatrixAlgebraKit.default_lq_algorithm(::Type{T}; kwargs...) where {T <:
     return MatrixAlgebraKit.LQViaTransposedQR(GLA_HouseholderQR(; kwargs...))
 end
 
+MatrixAlgebraKit.left_orth_alg(alg::GLA_HouseholderQR) = MatrixAlgebraKit.LeftOrthViaQR(alg)
+
 end
diff --git a/test/linearmap.jl b/test/linearmap.jl
index ad6102d7..a7adaae7 100644
--- a/test/linearmap.jl
+++ b/test/linearmap.jl
@@ -3,7 +3,8 @@ module LinearMaps
     export LinearMap
 
     using MatrixAlgebraKit
-    using MatrixAlgebraKit: AbstractAlgorithm, DiagonalAlgorithm
+    using MatrixAlgebraKit: AbstractAlgorithm, DiagonalAlgorithm, GLA_QRIteration
+    using GenericLinearAlgebra
     import MatrixAlgebraKit as MAK
 
     using LinearAlgebra: LinearAlgebra, lmul!, rmul!
@@ -30,8 +31,12 @@ module LinearMaps
             MAK.check_input($f!, parent(A), parent.(F), alg)
         @eval MAK.initialize_output(::typeof($f!), A::LinearMap, alg::AbstractAlgorithm) =
             LinearMap.(MAK.initialize_output($f!, parent(A), alg))
+        @eval MAK.initialize_output(::typeof($f!), A::LinearMap, alg::GLA_QRIteration) =
+            (nothing, nothing, nothing)
         @eval MAK.$f!(A::LinearMap, F, alg::AbstractAlgorithm) =
             LinearMap.(MAK.$f!(parent(A), parent.(F), alg))
+        @eval MAK.$f!(A::LinearMap, F, alg::GLA_QRIteration) =
+            LinearMap.(MAK.$f!(parent(A), F, alg))
         @eval MAK.check_input(::typeof($f!), A::LinearMap, F, alg::DiagonalAlgorithm) =
             MAK.check_input($f!, parent(A), parent.(F), alg)
         @eval MAK.initialize_output(::typeof($f!), A::LinearMap, alg::DiagonalAlgorithm) =
diff --git a/test/orthnull.jl b/test/orthnull.jl
index 5db9d2de..f2e0e611 100644
--- a/test/orthnull.jl
+++ b/test/orthnull.jl
@@ -6,7 +6,7 @@ using LinearAlgebra: LinearAlgebra, I, Diagonal
 using CUDA, AMDGPU
 
 BLASFloats = (Float32, Float64, ComplexF32, ComplexF64)
-GenericFloats = (Float16, BigFloat, Complex{BigFloat})
+GenericFloats = (BigFloat, Complex{BigFloat})
 
 @isdefined(TestSuite) || include("testsuite/TestSuite.jl")
 using .TestSuite
@@ -27,9 +27,7 @@ for T in (BLASFloats..., GenericFloats...), n in (37, m, 63)
         end
     end
     if !is_buildkite
-        if T ∈ BLASFloats # no qr_null or lq_null for GenericFloats
-            TestSuite.test_orthnull(T, (m, n))
-        end
+        TestSuite.test_orthnull(T, (m, n))
         #AT = Diagonal{T, Vector{T}}
         #TestSuite.test_orthnull(AT, m) # not supported
     end

From dd09fe22f77fb2b3470d6c18d7a1f75609614714 Mon Sep 17 00:00:00 2001
From: Katharine Hyatt <katharine.s.hyatt@gmail.com>
Date: Wed, 31 Dec 2025 11:04:14 +0100
Subject: [PATCH 6/8] Try to fix testinferred with algs

---
 test/testsuite/orthnull.jl | 62 +++++++++++++++++++++-----------------
 1 file changed, 35 insertions(+), 27 deletions(-)

diff --git a/test/testsuite/orthnull.jl b/test/testsuite/orthnull.jl
index fa09214e..4a3c7b0e 100644
--- a/test/testsuite/orthnull.jl
+++ b/test/testsuite/orthnull.jl
@@ -3,6 +3,20 @@ using LinearAlgebra
 
 include("../linearmap.jl")
 
+_left_orth_svd(x; kwargs...) = left_orth(x; alg = :svd, kwargs...)
+_left_orth_svd!(x, VC; kwargs...) = left_orth!(x, VC; alg = :svd, kwargs...)
+_left_orth_qr(x; kwargs...) = left_orth(x; alg = :qr, kwargs...)
+_left_orth_qr!(x, VC; kwargs...) = left_orth!(x, VC; alg = :qr, kwargs...)
+_left_orth_polar(x; kwargs...) = left_orth(x; alg = :polar, kwargs...)
+_left_orth_polar!(x, VC; kwargs...) = left_orth!(x, VC; alg = :polar, kwargs...)
+
+_right_orth_svd(x; kwargs...) = right_orth(x; alg = :svd, kwargs...)
+_right_orth_svd!(x, CVᴴ; kwargs...) = right_orth!(x, CVᴴ; alg = :svd, kwargs...)
+_right_orth_lq(x; kwargs...) = right_orth(x; alg = :lq, kwargs...)
+_right_orth_lq!(x, CVᴴ; kwargs...) = right_orth!(x, CVᴴ; alg = :lq, kwargs...)
+_right_orth_polar(x; kwargs...) = right_orth(x; alg = :polar, kwargs...)
+_right_orth_polar!(x, CVᴴ; kwargs...) = right_orth!(x, CVᴴ; alg = :polar, kwargs...)
+
 function test_orthnull(T::Type, sz; kwargs...)
     summary_str = testargs_summary(T, sz)
     return @testset "orthnull $summary_str" begin
@@ -34,9 +48,7 @@ function test_left_orthnull(
         @test collect(V) * collect(V)' + collect(N) * collect(N)' ≈ I
 
         M = LinearMap(A)
-        # broken
-        #VM, CM = @testinferred left_orth(M; alg = :svd)
-        VM, CM = left_orth(M; alg = :svd)
+        VM, CM = @testinferred _left_orth_svd(M)
         @test parent(VM) * parent(CM) ≈ A
 
         if m > n && (T <: Number  || T <: Diagonal{<:Number, <:Vector})
@@ -53,9 +65,7 @@ function test_left_orthnull(
         end
 
         # passing a kind and some kwargs
-        # broken
-        # V, C = @testinferred left_orth(A; alg = :qr, positive = true)
-        V, C = left_orth(A; alg = :qr, positive = true)
+        V, C = @testinferred _left_orth_qr(A; positive = true)
         N = @testinferred left_null(A; alg = :qr, positive = true)
         @test V isa typeof(A) && size(V) == (m, minmn)
         @test C isa typeof(A) && size(C) == (minmn, n)
@@ -117,9 +127,13 @@ function test_left_orthnull(
 
         for alg in (:qr, :polar, :svd) # explicit kind kwarg
             m < n && alg === :polar && continue
-            # broken
-            # V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C); alg = alg)
-            V2, C2 = left_orth!(copy!(Ac, A), (V, C); alg = alg)
+            if alg == :svd
+                V2, C2 = @testinferred _left_orth_svd!(copy!(Ac, A), (V, C))
+            elseif alg == :qr
+                V2, C2 = @testinferred _left_orth_qr!(copy!(Ac, A), (V, C))
+            elseif alg == :polar
+                V2, C2 = @testinferred _left_orth_polar!(copy!(Ac, A), (V, C))
+            end
             @test V2 * C2 ≈ A
             @test isisometric(V2)
             if alg != :polar
@@ -132,9 +146,7 @@ function test_left_orthnull(
             # with kind and tol kwargs
             if alg == :svd
                 if (T <: Number  || T <: Diagonal{<:Number, <:Vector})
-                    # broken
-                    # V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C); alg = alg, trunc = (; atol))
-                    V2, C2 = left_orth!(copy!(Ac, A), (V, C); alg = alg, trunc = (; atol))
+                    V2, C2 = @testinferred _left_orth_svd!(copy!(Ac, A), (V, C); trunc = (; atol))
                     N2 = @testinferred left_null!(copy!(Ac, A), N; alg, trunc = (; atol))
                     @test V2 * C2 ≈ A
                     @test isisometric(V2)
@@ -142,9 +154,7 @@ function test_left_orthnull(
                     @test isisometric(N2)
                     @test collect(V2) * collect(V2)' + collect(N2) * collect(N2)' ≈ I
 
-                    # broken
-                    # V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C); alg = alg, trunc = (; rtol))
-                    V2, C2 = left_orth!(copy!(Ac, A), (V, C); alg = alg, trunc = (; rtol))
+                    V2, C2 = @testinferred _left_orth_svd!(copy!(Ac, A), (V, C); trunc = (; rtol))
                     N2 = @testinferred left_null!(copy!(Ac, A), N; alg, trunc = (; rtol))
                     @test V2 * C2 ≈ A
                     @test isisometric(V2)
@@ -186,9 +196,7 @@ function test_right_orthnull(
         @test collect(Vᴴ)' * collect(Vᴴ) + collect(Nᴴ)' * collect(Nᴴ) ≈ I
 
         M = LinearMap(A)
-        # broken
-        #CM, VMᴴ = @testinferred right_orth(M; alg = :svd)
-        CM, VMᴴ = right_orth(M; alg = :svd)
+        CM, VMᴴ = @testinferred _right_orth_svd(M)
         @test parent(CM) * parent(VMᴴ) ≈ A
 
         Ac = similar(A)
@@ -222,9 +230,13 @@ function test_right_orthnull(
 
         for alg in (:lq, :polar, :svd)
             n < m && alg == :polar && continue
-            # broken
-            #C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = alg)
-            C2, Vᴴ2 = right_orth!(copy!(Ac, A), (C, Vᴴ); alg = alg)
+            if alg == :lq
+                C2, Vᴴ2 = @testinferred _right_orth_lq!(copy!(Ac, A), (C, Vᴴ))
+            elseif alg == :polar
+                C2, Vᴴ2 = @testinferred _right_orth_polar!(copy!(Ac, A), (C, Vᴴ))
+            elseif alg == :svd
+                C2, Vᴴ2 = @testinferred _right_orth_svd!(copy!(Ac, A), (C, Vᴴ))
+            end
             @test C2 * Vᴴ2 ≈ A
             @test isisometric(Vᴴ2; side = :right)
             if alg != :polar
@@ -236,9 +248,7 @@ function test_right_orthnull(
 
             if alg == :svd
                 if (T <: Number  || T <: Diagonal{<:Number, <:Vector})
-                    # broken
-                    #C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = alg, trunc = (; atol))
-                    C2, Vᴴ2 = right_orth!(copy!(Ac, A), (C, Vᴴ); alg = alg, trunc = (; atol))
+                    C2, Vᴴ2 = @testinferred _right_orth_svd!(copy!(Ac, A), (C, Vᴴ); trunc = (; atol))
                     Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; alg = alg, trunc = (; atol))
                     @test C2 * Vᴴ2 ≈ A
                     @test isisometric(Vᴴ2; side = :right)
@@ -246,9 +256,7 @@ function test_right_orthnull(
                     @test isisometric(Nᴴ2; side = :right)
                     @test collect(Vᴴ2)' * collect(Vᴴ2) + collect(Nᴴ2)' * collect(Nᴴ2) ≈ I
 
-                    # broken
-                    #C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ); alg = alg, trunc = (; rtol))
-                    C2, Vᴴ2 = right_orth!(copy!(Ac, A), (C, Vᴴ); alg = alg, trunc = (; rtol))
+                    C2, Vᴴ2 = @testinferred _right_orth_svd!(copy!(Ac, A), (C, Vᴴ); trunc = (; rtol))
                     Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; alg = alg, trunc = (; rtol))
                     @test C2 * Vᴴ2 ≈ A
                     @test isisometric(Vᴴ2; side = :right)

From 04d929e783896bb1a9d7a0f1f27fc5f39397db49 Mon Sep 17 00:00:00 2001
From: Katharine Hyatt <kslimes@gmail.com>
Date: Mon, 5 Jan 2026 14:07:45 +0100
Subject: [PATCH 7/8] Split up tests and address other comments

---
 src/interface/orthnull.jl   |   2 +
 test/orthnull.jl            |  12 +-
 test/testsuite/TestSuite.jl |   9 +-
 test/testsuite/orthnull.jl  | 296 +++++++++++++++++++++---------------
 4 files changed, 188 insertions(+), 131 deletions(-)

diff --git a/src/interface/orthnull.jl b/src/interface/orthnull.jl
index 64acb509..301aef8a 100644
--- a/src/interface/orthnull.jl
+++ b/src/interface/orthnull.jl
@@ -443,6 +443,7 @@ left_orth_alg(alg::LeftOrthAlgorithm) = alg
 left_orth_alg(alg::QRAlgorithms) = LeftOrthViaQR(alg)
 left_orth_alg(alg::PolarAlgorithms) = LeftOrthViaPolar(alg)
 left_orth_alg(alg::SVDAlgorithms) = LeftOrthViaSVD(alg)
+left_orth_alg(alg::DiagonalAlgorithm) = LeftOrthViaSVD(alg)
 left_orth_alg(alg::TruncatedAlgorithm{<:SVDAlgorithms}) = LeftOrthViaSVD(alg)
 
 """
@@ -478,6 +479,7 @@ right_orth_alg(alg::RightOrthAlgorithm) = alg
 right_orth_alg(alg::LQAlgorithms) = RightOrthViaLQ(alg)
 right_orth_alg(alg::PolarAlgorithms) = RightOrthViaPolar(alg)
 right_orth_alg(alg::SVDAlgorithms) = RightOrthViaSVD(alg)
+right_orth_alg(alg::DiagonalAlgorithm) = RightOrthViaSVD(alg)
 right_orth_alg(alg::TruncatedAlgorithm{<:SVDAlgorithms}) = RightOrthViaSVD(alg)
 
 """
diff --git a/test/orthnull.jl b/test/orthnull.jl
index f2e0e611..dec12946 100644
--- a/test/orthnull.jl
+++ b/test/orthnull.jl
@@ -18,17 +18,17 @@ for T in (BLASFloats..., GenericFloats...), n in (37, m, 63)
     TestSuite.seed_rng!(123)
     if T ∈ BLASFloats
         if CUDA.functional()
-            TestSuite.test_orthnull(CuMatrix{T}, (m, n))
-            n == m && TestSuite.test_orthnull(Diagonal{T, CuVector{T}}, m)
+            TestSuite.test_orthnull(CuMatrix{T}, (m, n); test_nullity = false)
+            n == m && TestSuite.test_orthnull(Diagonal{T, CuVector{T}}, m; test_orthnull = false)
         end
         if AMDGPU.functional()
-            TestSuite.test_orthnull(ROCMatrix{T}, (m, n))
-            n == m && TestSuite.test_orthnull(Diagonal{T, ROCVector{T}}, m)
+            TestSuite.test_orthnull(ROCMatrix{T}, (m, n); test_nullity = false)
+            n == m && TestSuite.test_orthnull(Diagonal{T, ROCVector{T}}, m; test_orthnull = false)
         end
     end
     if !is_buildkite
         TestSuite.test_orthnull(T, (m, n))
-        #AT = Diagonal{T, Vector{T}}
-        #TestSuite.test_orthnull(AT, m) # not supported
+        AT = Diagonal{T, Vector{T}}
+        TestSuite.test_orthnull(AT, m; test_orthnull = false)
     end
 end
diff --git a/test/testsuite/TestSuite.jl b/test/testsuite/TestSuite.jl
index f288eff5..734d48fd 100644
--- a/test/testsuite/TestSuite.jl
+++ b/test/testsuite/TestSuite.jl
@@ -11,7 +11,7 @@ module TestSuite
 using Test
 using MatrixAlgebraKit
 using MatrixAlgebraKit: diagview
-using LinearAlgebra: Diagonal, norm, istriu, istril
+using LinearAlgebra: Diagonal, norm, istriu, istril, I
 using Random, StableRNGs
 using AMDGPU, CUDA
 
@@ -69,6 +69,13 @@ is_positive(alg::MatrixAlgebraKit.ROCSOLVER_HouseholderQR) = alg.positive
 is_positive(alg::MatrixAlgebraKit.LQViaTransposedQR) = is_positive(alg.qr_alg)
 is_pivoted(alg::MatrixAlgebraKit.LQViaTransposedQR) = is_pivoted(alg.qr_alg)
 
+isleftcomplete(V, N) = V * V' + N * N' ≈ I
+isleftcomplete(V::AnyCuMatrix, N::AnyCuMatrix) = isleftcomplete(collect(V), collect(N))
+isleftcomplete(V::AnyROCMatrix, N::AnyROCMatrix) = isleftcomplete(collect(V), collect(N))
+isrightcomplete(Vᴴ, Nᴴ) = Vᴴ' * Vᴴ + Nᴴ' * Nᴴ ≈ I
+isrightcomplete(V::AnyCuMatrix, N::AnyCuMatrix) = isrightcomplete(collect(V), collect(N))
+isrightcomplete(V::AnyROCMatrix, N::AnyROCMatrix) = isrightcomplete(collect(V), collect(N))
+
 include("qr.jl")
 include("lq.jl")
 include("polar.jl")
diff --git a/test/testsuite/orthnull.jl b/test/testsuite/orthnull.jl
index 4a3c7b0e..79349d2a 100644
--- a/test/testsuite/orthnull.jl
+++ b/test/testsuite/orthnull.jl
@@ -17,11 +17,83 @@ _right_orth_lq!(x, CVᴴ; kwargs...) = right_orth!(x, CVᴴ; alg = :lq, kwargs..
 _right_orth_polar(x; kwargs...) = right_orth(x; alg = :polar, kwargs...)
 _right_orth_polar!(x, CVᴴ; kwargs...) = right_orth!(x, CVᴴ; alg = :polar, kwargs...)
 
-function test_orthnull(T::Type, sz; kwargs...)
+function test_orthnull(T::Type, sz; test_nullity = true, test_orthnull = true, kwargs...)
     summary_str = testargs_summary(T, sz)
     return @testset "orthnull $summary_str" begin
-        test_left_orthnull(T, sz; kwargs...)
-        test_right_orthnull(T, sz; kwargs...)
+        test_orthnull && test_left_orthnull(T, sz; kwargs...)
+        test_nullity  && test_left_nullity(T, sz; kwargs...)
+        test_orthnull && test_right_orthnull(T, sz; kwargs...)
+        test_nullity  && test_right_nullity(T, sz; kwargs...)
+    end
+end
+
+function test_left_nullity(
+        T::Type, sz;
+        atol::Real = 0, rtol::Real = precision(T),
+        kwargs...
+    )
+    summary_str = testargs_summary(T, sz)
+    return @testset "left_nullity $summary_str" begin
+        A = instantiate_matrix(T, sz)
+        Ac = deepcopy(A)
+        m, n = size(A)
+        minmn = min(m, n)
+        if m > n
+            nullity = 5
+            V, C = @testinferred left_orth(A)
+            N = @testinferred left_null(A; trunc = (; maxnullity = nullity))
+            @test V isa typeof(A) && size(V) == (m, minmn)
+            @test C isa typeof(A) && size(C) == (minmn, n)
+            @test eltype(N) == eltype(A) && size(N) == (m, nullity)
+            @test V * C ≈ A
+            @test isisometric(V)
+            @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(eltype(T))
+            @test isisometric(N)
+        end
+
+        rtol = eps(real(eltype(T)))
+        for (trunc_orth, trunc_null) in (
+                ((; rtol = rtol), (; rtol = rtol)),
+                (trunctol(; rtol), trunctol(; rtol, keep_below = true)),
+            )
+            V, C = left_orth(A)
+            N = left_null(A)
+            V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C); trunc = trunc_orth)
+            N2 = @testinferred left_null!(copy!(Ac, A), N; trunc = trunc_null)
+            @test V2 * C2 ≈ A
+            @test isisometric(V2)
+            @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(eltype(T))
+            @test isisometric(N2)
+            @test isleftcomplete(V2, N2)
+        end
+
+        alg = :svd
+        V2, C2 = @testinferred _left_orth_svd!(copy!(Ac, A), (V, C); trunc = (; atol))
+        N2 = @testinferred left_null!(copy!(Ac, A), N; alg, trunc = (; atol))
+        @test V2 * C2 ≈ A
+        @test isisometric(V2)
+        @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(eltype(T))
+        @test isisometric(N2)
+        @test isleftcomplete(V2, N2)
+
+        V2, C2 = @testinferred _left_orth_svd!(copy!(Ac, A), (V, C); trunc = (; rtol))
+        N2 = @testinferred left_null!(copy!(Ac, A), N; alg, trunc = (; rtol))
+        @test V2 * C2 ≈ A
+        @test isisometric(V2)
+        @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(eltype(T))
+        @test isisometric(N2)
+        @test isleftcomplete(V2, N2)
+
+        # doesn't work on AMD...
+        atol = eps(real(eltype(T)))
+        V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C); trunc = (; atol = atol))
+        N2 = @testinferred left_null!(copy!(Ac, A), N; trunc = (; atol = atol))
+        @test V2 * C2 ≈ A
+        @test isisometric(V2)
+        @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+        @test isisometric(N2)
+        @test isleftcomplete(V2, N2)
+
     end
 end
 
@@ -45,25 +117,12 @@ function test_left_orthnull(
         @test isisometric(V)
         @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
         @test isisometric(N)
-        @test collect(V) * collect(V)' + collect(N) * collect(N)' ≈ I
+        @test isleftcomplete(V, N)
 
         M = LinearMap(A)
         VM, CM = @testinferred _left_orth_svd(M)
         @test parent(VM) * parent(CM) ≈ A
 
-        if m > n && (T <: Number  || T <: Diagonal{<:Number, <:Vector})
-            nullity = 5
-            V, C = @testinferred left_orth(A)
-            N = @testinferred left_null(A; trunc = (; maxnullity = nullity))
-            @test V isa typeof(A) && size(V) == (m, minmn)
-            @test C isa typeof(A) && size(C) == (minmn, n)
-            @test eltype(N) == eltype(A) && size(N) == (m, nullity)
-            @test V * C ≈ A
-            @test isisometric(V)
-            @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-            @test isisometric(N)
-        end
-
         # passing a kind and some kwargs
         V, C = @testinferred _left_orth_qr(A; positive = true)
         N = @testinferred left_null(A; alg = :qr, positive = true)
@@ -74,56 +133,27 @@ function test_left_orthnull(
         @test isisometric(V)
         @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
         @test isisometric(N)
-        @test collect(V) * collect(V)' + collect(N) * collect(N)' ≈ I
+        @test isleftcomplete(V, N)
 
         # passing an algorithm
-        if !isa(A, Diagonal)
-            V, C = @testinferred left_orth(A; alg = MatrixAlgebraKit.default_qr_algorithm(A))
-            N = @testinferred left_null(A; alg = :qr, positive = true)
-            @test V isa typeof(A) && size(V) == (m, minmn)
-            @test C isa typeof(A) && size(C) == (minmn, n)
-            @test eltype(N) == eltype(A) && size(N) == (m, m - minmn)
-            @test V * C ≈ A
-            @test isisometric(V)
-            @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-            @test isisometric(N)
-            @test collect(V) * collect(V)' + collect(N) * collect(N)' ≈ I
-        end
+        V, C = @testinferred left_orth(A; alg = MatrixAlgebraKit.default_qr_algorithm(A))
+        N = @testinferred left_null(A; alg = :qr, positive = true)
+        @test V isa typeof(A) && size(V) == (m, minmn)
+        @test C isa typeof(A) && size(C) == (minmn, n)
+        @test eltype(N) == eltype(A) && size(N) == (m, m - minmn)
+        @test V * C ≈ A
+        @test isisometric(V)
+        @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+        @test isisometric(N)
+        @test isleftcomplete(V, N)
 
-        Ac = similar(A)
         V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C))
         N2 = @testinferred left_null!(copy!(Ac, A), N)
         @test V2 * C2 ≈ A
         @test isisometric(V2)
         @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
         @test isisometric(N2)
-        @test collect(V2) * collect(V2)' + collect(N2) * collect(N2)' ≈ I
-
-        # doesn't work on AMD...
-        atol = eps(real(eltype(T)))
-        V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C); trunc = (; atol = atol))
-        N2 = @testinferred left_null!(copy!(Ac, A), N; trunc = (; atol = atol))
-        @test V2 * C2 ≈ A
-        @test isisometric(V2)
-        @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometric(N2)
-        @test collect(V2) * collect(V2)' + collect(N2) * collect(N2)' ≈ I
-
-        if (T <: Number  || T <: Diagonal{<:Number, <:Vector})
-            rtol = eps(real(eltype(T)))
-            for (trunc_orth, trunc_null) in (
-                    ((; rtol = rtol), (; rtol = rtol)),
-                    (trunctol(; rtol), trunctol(; rtol, keep_below = true)),
-                )
-                V2, C2 = @testinferred left_orth!(copy!(Ac, A), (V, C); trunc = trunc_orth)
-                N2 = @testinferred left_null!(copy!(Ac, A), N; trunc = trunc_null)
-                @test V2 * C2 ≈ A
-                @test isisometric(V2)
-                @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometric(N2)
-                @test collect(V2) * collect(V2)' + collect(N2) * collect(N2)' ≈ I
-            end
-        end
+        @test isleftcomplete(V2, N2)
 
         for alg in (:qr, :polar, :svd) # explicit kind kwarg
             m < n && alg === :polar && continue
@@ -140,29 +170,11 @@ function test_left_orthnull(
                 N2 = @testinferred left_null!(copy!(Ac, A), N; alg)
                 @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
                 @test isisometric(N2)
-                @test collect(V2) * collect(V2)' + collect(N2) * collect(N2)' ≈ I
+                @test isleftcomplete(V2, N2)
             end
 
             # with kind and tol kwargs
-            if alg == :svd
-                if (T <: Number  || T <: Diagonal{<:Number, <:Vector})
-                    V2, C2 = @testinferred _left_orth_svd!(copy!(Ac, A), (V, C); trunc = (; atol))
-                    N2 = @testinferred left_null!(copy!(Ac, A), N; alg, trunc = (; atol))
-                    @test V2 * C2 ≈ A
-                    @test isisometric(V2)
-                    @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                    @test isisometric(N2)
-                    @test collect(V2) * collect(V2)' + collect(N2) * collect(N2)' ≈ I
-
-                    V2, C2 = @testinferred _left_orth_svd!(copy!(Ac, A), (V, C); trunc = (; rtol))
-                    N2 = @testinferred left_null!(copy!(Ac, A), N; alg, trunc = (; rtol))
-                    @test V2 * C2 ≈ A
-                    @test isisometric(V2)
-                    @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                    @test isisometric(N2)
-                    @test collect(V2) * collect(V2)' + collect(N2) * collect(N2)' ≈ I
-                end
-            else
+            if alg != :svd
                 @test_throws ArgumentError left_orth!(copy!(Ac, A), (V, C); alg, trunc = (; atol))
                 @test_throws ArgumentError left_orth!(copy!(Ac, A), (V, C); alg, trunc = (; rtol))
                 alg == :polar && continue
@@ -173,6 +185,57 @@ function test_left_orthnull(
     end
 end
 
+function test_right_nullity(
+        T::Type, sz;
+        atol::Real = 0, rtol::Real = precision(T),
+        kwargs...
+    )
+    summary_str = testargs_summary(T, sz)
+    return @testset "right_nullity $summary_str" begin
+        A = instantiate_matrix(T, sz)
+        Ac = deepcopy(A)
+        m, n = size(A)
+        minmn = min(m, n)
+
+        C, Vᴴ = @testinferred right_orth(A)
+        Nᴴ = @testinferred right_null(A)
+        atol = eps(real(eltype(T)))
+        C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ); trunc = (; atol))
+        Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; trunc = (; atol))
+        @test C2 * Vᴴ2 ≈ A
+        @test isisometric(Vᴴ2; side = :right)
+        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(eltype(T))
+        @test isisometric(Nᴴ; side = :right)
+        @test isrightcomplete(Vᴴ2, Nᴴ2)
+
+        rtol = eps(real(eltype(T)))
+        C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ); trunc = (; rtol))
+        Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; trunc = (; rtol))
+        @test C2 * Vᴴ2 ≈ A
+        @test isisometric(Vᴴ2; side = :right)
+        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(eltype(T))
+        @test isisometric(Nᴴ2; side = :right)
+        @test isrightcomplete(Vᴴ2, Nᴴ2)
+
+        alg = :svd
+        C2, Vᴴ2 = @testinferred _right_orth_svd!(copy!(Ac, A), (C, Vᴴ); trunc = (; atol))
+        Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; alg = alg, trunc = (; atol))
+        @test C2 * Vᴴ2 ≈ A
+        @test isisometric(Vᴴ2; side = :right)
+        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+        @test isisometric(Nᴴ2; side = :right)
+        @test isrightcomplete(Vᴴ2, Nᴴ2)
+
+        C2, Vᴴ2 = @testinferred _right_orth_svd!(copy!(Ac, A), (C, Vᴴ); trunc = (; rtol))
+        Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; alg = alg, trunc = (; rtol))
+        @test C2 * Vᴴ2 ≈ A
+        @test isisometric(Vᴴ2; side = :right)
+        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+        @test isisometric(Nᴴ2; side = :right)
+        @test isrightcomplete(Vᴴ2, Nᴴ2)
+    end
+end
+
 function test_right_orthnull(
         T::Type, sz;
         atol::Real = 0, rtol::Real = precision(T),
@@ -191,42 +254,45 @@ function test_right_orthnull(
         @test eltype(Nᴴ) == eltype(A) && size(Nᴴ) == (n - minmn, n)
         @test C * Vᴴ ≈ A
         @test isisometric(Vᴴ; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+        @test LinearAlgebra.norm(A * adjoint(Nᴴ)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(eltype(T))
         @test isisometric(Nᴴ; side = :right)
-        @test collect(Vᴴ)' * collect(Vᴴ) + collect(Nᴴ)' * collect(Nᴴ) ≈ I
+        @test isrightcomplete(Vᴴ, Nᴴ)
 
         M = LinearMap(A)
         CM, VMᴴ = @testinferred _right_orth_svd(M)
         @test parent(CM) * parent(VMᴴ) ≈ A
 
-        Ac = similar(A)
+        # passing a kind and some kwargs
+        C, Vᴴ = @testinferred _right_orth_lq(A; positive = true)
+        Nᴴ = @testinferred right_null(A; alg = :lq, positive = true)
+        @test C isa typeof(A) && size(C) == (m, minmn)
+        @test Vᴴ isa typeof(A) && size(Vᴴ) == (minmn, n)
+        @test eltype(Nᴴ) == eltype(A) && size(Nᴴ) == (n - minmn, n)
+        @test C * Vᴴ ≈ A
+        @test isisometric(Vᴴ; side = :right)
+        @test LinearAlgebra.norm(A * adjoint(Nᴴ)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(eltype(T))
+        @test isisometric(Nᴴ; side = :right)
+        @test isrightcomplete(Vᴴ, Nᴴ)
+
+        # passing an algorithm
+        C, Vᴴ = @testinferred right_orth(A; alg = MatrixAlgebraKit.default_lq_algorithm(A))
+        Nᴴ = @testinferred right_null(A; alg = :lq, positive = true)
+        @test C isa typeof(A) && size(C) == (m, minmn)
+        @test Vᴴ isa typeof(A) && size(Vᴴ) == (minmn, n)
+        @test eltype(Nᴴ) == eltype(A) && size(Nᴴ) == (n - minmn, n)
+        @test C * Vᴴ ≈ A
+        @test isisometric(Vᴴ; side = :right)
+        @test LinearAlgebra.norm(A * adjoint(Nᴴ)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(eltype(T))
+        @test isisometric(Nᴴ; side = :right)
+        @test isrightcomplete(Vᴴ, Nᴴ)
+
         C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ))
         Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ)
         @test C2 * Vᴴ2 ≈ A
         @test isisometric(Vᴴ2; side = :right)
-        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+        @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(eltype(T))
         @test isisometric(Nᴴ; side = :right)
-        @test collect(Vᴴ2)' * collect(Vᴴ2) + collect(Nᴴ2)' * collect(Nᴴ2) ≈ I
-
-        if (T <: Number  || T <: Diagonal{<:Number, <:Vector})
-            atol = eps(real(eltype(T)))
-            C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ); trunc = (; atol))
-            Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; trunc = (; atol))
-            @test C2 * Vᴴ2 ≈ A
-            @test isisometric(Vᴴ2; side = :right)
-            @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-            @test isisometric(Nᴴ; side = :right)
-            @test collect(Vᴴ2)' * collect(Vᴴ2) + collect(Nᴴ2)' * collect(Nᴴ2) ≈ I
-
-            rtol = eps(real(eltype(T)))
-            C2, Vᴴ2 = @testinferred right_orth!(copy!(Ac, A), (C, Vᴴ); trunc = (; rtol))
-            Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; trunc = (; rtol))
-            @test C2 * Vᴴ2 ≈ A
-            @test isisometric(Vᴴ2; side = :right)
-            @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-            @test isisometric(Nᴴ2; side = :right)
-            @test collect(Vᴴ2)' * collect(Vᴴ2) + collect(Nᴴ2)' * collect(Nᴴ2) ≈ I
-        end
+        @test isrightcomplete(Vᴴ2, Nᴴ2)
 
         for alg in (:lq, :polar, :svd)
             n < m && alg == :polar && continue
@@ -241,30 +307,12 @@ function test_right_orthnull(
             @test isisometric(Vᴴ2; side = :right)
             if alg != :polar
                 Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; alg = alg)
-                @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
+                @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(eltype(T))
                 @test isisometric(Nᴴ2; side = :right)
-                @test collect(Vᴴ2)' * collect(Vᴴ2) + collect(Nᴴ2)' * collect(Nᴴ2) ≈ I
+                @test isrightcomplete(Vᴴ2, Nᴴ2)
             end
 
-            if alg == :svd
-                if (T <: Number  || T <: Diagonal{<:Number, <:Vector})
-                    C2, Vᴴ2 = @testinferred _right_orth_svd!(copy!(Ac, A), (C, Vᴴ); trunc = (; atol))
-                    Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; alg = alg, trunc = (; atol))
-                    @test C2 * Vᴴ2 ≈ A
-                    @test isisometric(Vᴴ2; side = :right)
-                    @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                    @test isisometric(Nᴴ2; side = :right)
-                    @test collect(Vᴴ2)' * collect(Vᴴ2) + collect(Nᴴ2)' * collect(Nᴴ2) ≈ I
-
-                    C2, Vᴴ2 = @testinferred _right_orth_svd!(copy!(Ac, A), (C, Vᴴ); trunc = (; rtol))
-                    Nᴴ2 = @testinferred right_null!(copy!(Ac, A), Nᴴ; alg = alg, trunc = (; rtol))
-                    @test C2 * Vᴴ2 ≈ A
-                    @test isisometric(Vᴴ2; side = :right)
-                    @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                    @test isisometric(Nᴴ2; side = :right)
-                    @test collect(Vᴴ2)' * collect(Vᴴ2) + collect(Nᴴ2)' * collect(Nᴴ2) ≈ I
-                end
-            else
+            if alg != :svd
                 @test_throws ArgumentError right_orth!(copy!(Ac, A), (C, Vᴴ); alg, trunc = (; atol))
                 @test_throws ArgumentError right_orth!(copy!(Ac, A), (C, Vᴴ); alg, trunc = (; rtol))
                 alg == :polar && continue

From 9b450bd328177c6c86eb2d6d9a5c098142fa8df2 Mon Sep 17 00:00:00 2001
From: Katharine Hyatt <kslimes@gmail.com>
Date: Tue, 6 Jan 2026 10:19:40 +0100
Subject: [PATCH 8/8] Remove extraneous lines

---
 src/interface/orthnull.jl | 2 --
 1 file changed, 2 deletions(-)

diff --git a/src/interface/orthnull.jl b/src/interface/orthnull.jl
index 301aef8a..64acb509 100644
--- a/src/interface/orthnull.jl
+++ b/src/interface/orthnull.jl
@@ -443,7 +443,6 @@ left_orth_alg(alg::LeftOrthAlgorithm) = alg
 left_orth_alg(alg::QRAlgorithms) = LeftOrthViaQR(alg)
 left_orth_alg(alg::PolarAlgorithms) = LeftOrthViaPolar(alg)
 left_orth_alg(alg::SVDAlgorithms) = LeftOrthViaSVD(alg)
-left_orth_alg(alg::DiagonalAlgorithm) = LeftOrthViaSVD(alg)
 left_orth_alg(alg::TruncatedAlgorithm{<:SVDAlgorithms}) = LeftOrthViaSVD(alg)
 
 """
@@ -479,7 +478,6 @@ right_orth_alg(alg::RightOrthAlgorithm) = alg
 right_orth_alg(alg::LQAlgorithms) = RightOrthViaLQ(alg)
 right_orth_alg(alg::PolarAlgorithms) = RightOrthViaPolar(alg)
 right_orth_alg(alg::SVDAlgorithms) = RightOrthViaSVD(alg)
-right_orth_alg(alg::DiagonalAlgorithm) = RightOrthViaSVD(alg)
 right_orth_alg(alg::TruncatedAlgorithm{<:SVDAlgorithms}) = RightOrthViaSVD(alg)
 
 """